Download Electric Circuits I Midterm #1

The University of Toledo
EECS:2300 Electric Circuits
Dr. Anthony D. Johnson
Section number ________
Student Name _______________________________________
Electric Circuits I
Midterm #1
Problems
Points
1. _____
4
2. _____
6
3. _____
5
Total _____
15
Was the exam fair ?
yes
no
f11m1s_elci.fm - 1
The University of Toledo
EECS:2300 Electric Circuits
Dr. Anthony D. Johnson
Problem 1
Section number ________
f11m1s_elci.fm - 2
Student Name _______________________________________
4 points
Given is the electric circuit model, shown in Figure 1.1.
RV
b
VV = 24V
R1
+
VR1
-
VV
R2
Vab
IR1
RV = 10 kΩ
R1= 30 kΩ
R2= 20 kΩ
a
Fig.1.1 An electric circuit model and its parameter values.
For the electric circuit model of Figure 1.1, demonstrate:
1. an ability to apply the voltage divider formula, by using it to calculate the value of voltage
VR1 across the resistor R1 indicated in the circuit model of Figure 1.1;
2.
an ability to apply the Ohm’s Law, by using the Ohm’s Law to calculate the value of the
current IR1 flowing through resistor R1 in the circuit model of Figure 1.1.
Hint #1 For full credit: all equations, all answers to questions, all circuit models and other
graphical representations are expected to be entered into the space designated for them;
all shown numerical results must be preceded by the symbolic and numeric expressions
whose evaluation produces the numerical results.
Solution
For full credit, an explicit demonstration of understanding the following solution steps is expected.
1
1.1
Applying the voltage divider formula, calculate the value of the indicated voltage VR1 accross the
resistor R1 in the circuit model of Figure 1.1; show your work in the space reserved for equation
(1-1).
VR1 =
1
1.2
R1
RV + R2 + R1
VV =
30⋅103
24 = 12V
(10 + 20 + 30)⋅103
(1-1)
Applying the passive convention for coupled positive reference directions of the voltage and
current of the resistor R1, indicate in the circuit model of Figure 1.1 the positive reference direction
of the current IR1.
The University of Toledo
EECS:2300 Electric Circuits
Dr. Anthony D. Johnson
1
f11m1s_elci.fm - 3
Student Name _______________________________________
1.3 Using the positive reference direction determined for current IR1 in part 1.2 and applying the
Ohm’s Law, calculate the value of the current IR1 through resistor R1 in the circuit model of Figure
1.1; show your work in the space reserved for equation (1-2).
IR1 =
1
Section number ________
1.4
VR1
R1
=
12
= 0.4mA
30⋅103
(1-2)
Determine the positive reference direction, and the value of the magnitude, of the voltage Vab
between the terminals a and b in the circuit model of Figure 1.1; show the positive reference
direction of Vab in the circuit model of figure 1.1 and show your calculatiuon of the magnitude’s
value in the space reserved for equation (1-3).
Vab = R1⋅( −IR1) = 30⋅103 ⋅ ( −0.4⋅10-3) = - 12V
Or alternatively, using the calculated value of the voltage VR1,
Vab = - Vba = - VR1 = - 12V
(1-3)
The University of Toledo
EECS:2300 Electric Circuits
Dr. Anthony D. Johnson
Problem 2
Section number ________
f11m1s_elci.fm - 4
Student Name _______________________________________
6 points
Given is the electric circuit model, shown in Figure 2.1.
R 1= 4 Ω
R 3= 5 Ω
II = 15A
R 2= 2 Ω
R4= 10 Ω
VV = 25V
1
II
R1
R2
V1
2
R3
Vv
V2
-+
IVV
II
R4
R2
1
R1
2
R3
V1
V2
0
(a)
R4
IV
0
(b)
Figure 2.1 The electric circuit model with positive reference directions for currents and voltages that ought to be
calculated. (a)Drawing of the circuit model. (b)Equivalent representationof the circuit model if one will be used in the
solution process.
For the electric circuit model of Figure 2.1, demonstrate an ability to use the Nodal Voltage Method to
determine a partial solution that includes:
(a) voltage VI across the current source II,
(b) power delivered by the energy source VV to the circuit of Figure 2.1,
(c) electrical energy delivered by the energy source VV to the circuit of Figure 2.1 during a time
interval ∆t=5 minutes..
Hint #1 For full credit: all equations, all answers to questions, all circuit models and other
graphical representations are expected to be entered into the space designated for them;
all shown numerical results must be preceded by the symbolic and numeric expressions
whose evaluation produces the numerical results.
Solution
For full credit, an explicit demonstration of understanding the following solution steps is expected.
2.1
1
In any case, select and show in Figure 2.1 the positive reference directions for the nodal-voltages.
Additionally, when the solution processt, or just a simplification of the drawing, involves
replacement of a part of the circuita by an equivalent circuit, show the new equivalent form of the
whole electric circuit model in the space reserved for Figure 2.1(b); also write in the space reserved
for equation (2-1) any volttage-current relation needed to complete the electrical model shown in
Figure 2,1(b). .
As NVM is based on the application of the KCL, and a current-voltage relation does not exist for
an ideal voltage source, the application of NVM requires that part of the circuit consisting of the
series connection of the voltage source VV and resistor R4 be replaced by the equivalent Norton’s
circuit. After this replacement is applied, the circuit model of Figure 2.1(b) is obtained, and the
The University of Toledo
EECS:2300 Electric Circuits
Dr. Anthony D. Johnson
Section number ________
f11m1s_elci.fm - 5
Student Name _______________________________________
equation (2-1) explicitely defines the introduced Norton’s current source parameter IV .
IV = G4VV
0.5
2.2
(2-1)
For the circuit model of Figure 2.1(b), prepare the set of canonical form nodal-voltage equations.
Show your work in the space reserved for equations (2-2).
Since the circuit model of Figure 2.1(b) does not contain any voltage sources, direct application of
the NVM is applicable to it, and its corresponding normal form NVM system of equations is,
G11V1 - G12V2 = II
(2-2)
-G21V1 + G22V2 = - IV
1
2.3
Prepare the symbolic expressions and calculate the numerical values of the coefficients in
equations (2-2) (the self and mutual conductancess of the nodes); show the calculation in the space
reserved for equations (2-3).
G1 =
G2 =
G3 =
G4 =
2.4
1
∆=
1
R1
1
R2
1
R3
=
=
=
1
4
1
2
1
5
= 0.25 S
= 0.5 S
G11= G2 + G1 = 0.5 + 0.25 = 0.75 S
G22= G4 + G3 + G2 = 0.1 + 0.2 + 0.5 = 0.8 S
G12 = G21 = G2 = 0.5 S
= 0.2 S
(2-3)
1
1
R4
= 10 = 0.1 S
Prepare symbolic expressions (in terms of the nodal-voltage equation coefficients), and calculate
the values of determinants involved in the solution of equations (2-2); show the calculation in the
space reserved for equations (2-4).
G11
-G12
-G21
G22
= G11G22 -G21G12 = 0.75.0.8 - 0.5.0.5 = 0.35 S2
(2-4)
∆ 1=
∆ 2=
II
-G12
-G4VV
G22
G11
II
-G21 -G4VV
= G22 II - G4VV G12 = 0.8.15 -0.1.25.0.5 = 12 - 1.25= 10.75 SA
= - G4VV G11 + G21II = - 0.1.25.0.75 + 0.5.15= -1.875+7.5= 5.625 SA
The University of Toledo
EECS:2300 Electric Circuits
Dr. Anthony D. Johnson
0.5
2.5
2.6
1
Section number ________
f11m1s_elci.fm - 6
Student Name _______________________________________
Based on the values of determinants obtained in the step 2.4, calculate the numerical values of the
nodal-voltages; show the calculation in the space reserved for equations (2-5).
V1=
∆1
10.75
=
= 30.71 V
∆
0.35
V2 =
∆2
5.625
∆ = 0.35 = 16.07 V
(2-5)
Indicate in the circuit of Figure 2.1(a) the active convention positive reference direction for
the:current IVV flowing through the voltage source VV and then determine the value of current IVV;
show the calculation in the space reserved for equation (2-6).
Since there exists no current-voltage relation for an ideal voltage source, the currrent IVV can not
be determined directly; instead, as the voltage source VV and resistor R4 are connected in series,
their currents are equal, and determining the current IR4 of the resistor provides the value of the
current IVV too.
OL: IR4 = VR4⋅G4 = (VV + V2)⋅G4 = [25 + 16.07]⋅0.1 = 4.1A
IVV = IR4 = 4.1A
0.5
2.7
(2-6)
Calculate the power which the energy source VV delivers/consumes to/from the circuit of Figure
2.1; show the calculation in the space reserved for equations (2-7).
The power delivered by the energy sourceVV is
(2-7)
PV = VVIVV = 25⋅4.1 = 102.5 W
0.5
2.8
Calculate the amount of electrical energy WV deliverd/consumed by the energy source VV in the
circuit of Figure 1.1 during the time interval ∆t; show the calculation in the space reserved for
equations (2-8).
The energy deliverd/consumed by the energy source VV to the circuit during the time interval ∆t
is calculated as
WV = PV⋅∆t = 102.5⋅300 = 30.75 kJ ⇒ delivered
(2-8)
The University of Toledo
EECS:2300 Electric Circuits
Dr. Anthony D. Johnson
Problem 3
Section number ________
f11m1s_elci.fm - 7
Student Name _______________________________________
5 points
Given is a resistive network whose electrical model is shown in Figure 3.1.
a
R1
R2= 6 Ω
R5 = 20 Ω
R3 =60 Ω
R6 = 40 Ω
Rab
VV
R3
→
R5
R4
-
R4 = 20 Ω
+
R1 = 9 Ω
R2
b
VV = 25V
R6
Figure 3.1 Electrical model of a resistive network and its parametr values.
For the electric circuit model of Figure 3.1, demonstrate :
1. an ability to apply the series/parallel reduction method, to calculate the value of the
equivalent resistance Rab seen by the voltage source VV between terminals a and b,
2.
an ability to calculate the amount of power that the ideal voltage source VV delivers to the
resistive network in Figure 3.1.
Hint #1 For full credit: all equations, all answers to questions, all circuit models and other
graphical representations are expected to be entered into the space designated for them;
all shown numerical results must be preceded by the symbolic and numeric expressions
whose evaluation produces the numerical results.
Solution
For full credit, an explicit demonstration of understanding the following solution steps is expected
.3.1
2
For the resistive part of the network of Figure 3.1, prepare the graphical representations of three
equivalent networks of gradually decreasing complexity, which result when resistors connected in
series/parallel are replaced by an equivalent resistor. Show the three graphical representations, in
the order of their creation, in the space reserved for Figure 3.2. Label the equivalent resistances at
each reduction step by Rxy, where x and y are texts of your own choice.
a
R1
Rab →
R2
R34
b
R56
(a)
a
a
R1
Rab →
R2
Rab →
R1
b
b
R23456
R3456
(b)
Figure 3.2 Reduction steps for the resistive network of Figure 3.1.
(c)
The University of Toledo
EECS:2300 Electric Circuits
Dr. Anthony D. Johnson
3.2
2
Section number ________
f11m1s_elci.fm - 8
Student Name _______________________________________
Calculate the values of all equivalent resistances indicated in Figure 3.2; show the work in the
space reserved for equations (3-1).
R34 = R3|| R4 =
R3⋅R4
60⋅ 20
= 15Ω
R3+ R4 = 60 + 20
R56 = R5+ R6 = 20 + 40 = 60Ω
R3456 = R34|| R56 =
R34⋅R56
R34+ R56
=
15⋅ 60
= 12Ω
15 + 60
(3-1)
R23456 = R2+ R3456 = 6 + 12 = 18Ω
R1⋅R34567
R12 + R34567 =
Rab = R1|| R23456 =
3.3
1
9⋅ 18
9 + 18
= 6Ω
Calculate the power PV which the voltage source VV delivers to the resistive network in Figure 3.1.
Show the calculation in the space reserved for equation (3-2).
2
PV =
VV
=
Rab
25
6
2
= 104W
(3-2)